2,676 research outputs found
Lipchitzian Stability of Parametric Variational Inequalities Over Generalized Polyhedra in Banach Spaces
This paper concerns the study of solution maps to parameterized variational inequalities over generalized polyhedra in reflexive Banach spaces. It has been recognized that generalized polyhedral sets are significantly different from the usual convex polyhedra in infinite dimensions and play an important role in various applications to optimization, particularly to generalized linear programming. Our main goal is to fully characterize robust Lipschitzian stability of the aforementioned solutions maps entirely via their initial data. This is done on the base of the coderivative criterion in variational analysis via efficient calculations of the coderivative and related objects for the systems under consideration. The case of generalized polyhedra is essentially more involved in comparison with usual convex polyhedral sets and requires developing elaborated techniques and new proofs of variational analysis
Shape optimisation for a class of semilinear variational inequalities with applications to damage models
The present contribution investigates shape optimisation problems for a class
of semilinear elliptic variational inequalities with Neumann boundary
conditions. Sensitivity estimates and material derivatives are firstly derived
in an abstract operator setting where the operators are defined on polyhedral
subsets of reflexive Banach spaces. The results are then refined for
variational inequalities arising from minimisation problems for certain convex
energy functionals considered over upper obstacle sets in . One
particularity is that we allow for dynamic obstacle functions which may arise
from another optimisation problems. We prove a strong convergence property for
the material derivative and establish state-shape derivatives under regularity
assumptions. Finally, as a concrete application from continuum mechanics, we
show how the dynamic obstacle case can be used to treat shape optimisation
problems for time-discretised brittle damage models for elastic solids. We
derive a necessary optimality system for optimal shapes whose state variables
approximate desired damage patterns and/or displacement fields
Symmetric confidence regions and confidence intervals for normal map formulations of stochastic variational inequalities
Stochastic variational inequalities (SVI) model a large class of equilibrium
problems subject to data uncertainty, and are closely related to stochastic
optimization problems. The SVI solution is usually estimated by a solution to a
sample average approximation (SAA) problem. This paper considers the normal map
formulation of an SVI, and proposes a method to build asymptotically exact
confidence regions and confidence intervals for the solution of the normal map
formulation, based on the asymptotic distribution of SAA solutions. The
confidence regions are single ellipsoids with high probability. We also discuss
the computation of simultaneous and individual confidence intervals
Second-order subdifferential calculus with applications to tilt stability in optimization
The paper concerns the second-order generalized differentiation theory of
variational analysis and new applications of this theory to some problems of
constrained optimization in finitedimensional spaces. The main attention is
paid to the so-called (full and partial) second-order subdifferentials of
extended-real-valued functions, which are dual-type constructions generated by
coderivatives of frst-order subdifferential mappings. We develop an extended
second-order subdifferential calculus and analyze the basic second-order
qualification condition ensuring the fulfillment of the principal secondorder
chain rule for strongly and fully amenable compositions. The calculus results
obtained in this way and computing the second-order subdifferentials for
piecewise linear-quadratic functions and their major specifications are applied
then to the study of tilt stability of local minimizers for important classes
of problems in constrained optimization that include, in particular, problems
of nonlinear programming and certain classes of extended nonlinear programs
described in composite terms
Discrete conformal maps and ideal hyperbolic polyhedra
We establish a connection between two previously unrelated topics: a
particular discrete version of conformal geometry for triangulated surfaces,
and the geometry of ideal polyhedra in hyperbolic three-space. Two triangulated
surfaces are considered discretely conformally equivalent if the edge lengths
are related by scale factors associated with the vertices. This simple
definition leads to a surprisingly rich theory featuring M\"obius invariance,
the definition of discrete conformal maps as circumcircle preserving piecewise
projective maps, and two variational principles. We show how literally the same
theory can be reinterpreted to addresses the problem of constructing an ideal
hyperbolic polyhedron with prescribed intrinsic metric. This synthesis enables
us to derive a companion theory of discrete conformal maps for hyperbolic
triangulations. It also shows how the definitions of discrete conformality
considered here are closely related to the established definition of discrete
conformality in terms of circle packings.Comment: 62 pages, 22 figures. v2: typos corrected, references added and
updated, minor changes in exposition. v3, final version: typos corrected,
improved exposition, some material moved to appendice
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